Propositional Logic Normal Forms 1 Literals Definition A literal is an atom or the negation of an atom. In the former case the literal is positive, in the latter case it is negative. 2 Negation Normal Form (NNF) Definition A formula is in negation formal form (NNF) if negation (:) occurs only directly in front of atoms. Example In NNF: :A ^ :B Not in NNF: :(A _ B) 3 Transformation into NNF Any formula can be transformed into an equivalent formula in NNF by pushing : inwards. Apply the following equivalences from left to right as long as possible: ::F ≡ F :(F ^ G) ≡ (:F _:G) :(F _ G) ≡ (:F ^ :G) Example (:(A ^ :B) ^ C) ≡ ((:A _ ::B) ^ C) ≡ ((:A _ B) ^ C) Warning:\ F ≡ G ≡ H" is merely an abbreviation for \F ≡ G and G ≡ H" Does this process always terminate? Is the result unique? 4 CNF and DNF Definition A formula F is in conjunctive normal form (CNF) if it is a conjunction of disjunctions of literals: n m V Wi F = ( ( Li;j )), i=1 j=1 where Li;j 2 fA1; A2; · · · g [ f:A1; :A2; · · · g Definition A formula F is in disjunctive normal form (DNF) if it is a disjunction of conjunctions of literals: n m W Vi F = ( ( Li;j )), i=1 j=1 where Li;j 2 fA1; A2; · · · g [ f:A1; :A2; · · · g 5 Transformation into CNF and DNF Any formula can be transformed into an equivalent formula in CNF or DNF in two steps: 1. Transform the initial formula into its NNF 2. Transform the NNF into CNF or DNF: I Transformation into CNF. Apply the following equivalences from left to right as long as possible: (F _ (G ^ H)) ≡ ((F _ G) ^ (F _ H)) ((F ^ G) _ H) ≡ ((F _ H) ^ (G _ H)) I Transformation into DNF. Apply the following equivalences from left to right as long as possible: (F ^ (G _ H)) ≡ ((F ^ G) _ (F ^ H)) ((F _ G) ^ H) ≡ ((F ^ H) _ (G ^ H)) 6 Termination Why does the transformation into NNF and CNF terminate? Challenge Question: Find a weight function w :: formula ! N such that w(l:h:s:) > w(r:h:s:) for the equivalences ::F ≡ F :(F ^ G) ≡ (:F _:G) :(F _ G) ≡ (:F ^ :G) (F _ (G ^ H)) ≡ ((F _ G) ^ (F _ H)) ((F ^ G) _ H) ≡ ((F _ H) ^ (G _ H)) Define w recursively: w(Ai ) = ::: w(:F ) = ::: w(F ) ::: w(F ^ G) = ::: w(F ) ::: w(G) ::: w(F _ G) = ::: w(F ) ::: w(G) ::: 7 Complexity considerations The CNF and DNF of a formula of size n can have size 2n Can we do better? Yes, if we do not instist on ≡. Definition Two formulas F and G are equisatisfiable if F is satisfiable iff G is satisfiable. Theorem For every formula F of size n there is an equisatisfiable CNF formula G of size O(n). 8.